Question 969561
Determine the conic: 9x^2 + 2y^2 + 100y - 72x + 19 = 0, give its properties and sketch the graph.
***
9x^2- 72x + 2y^2 + 100y  + 19 = 0
complete the square:
9(x^2-8x+16) + 2(y^2+50y+625) =-19+144+1350
9(x-4)^2+2(y+25)^2=1475
equation of given ellipse:{{{(x-4)^2/(1475/9)+(y+25)^2/(1475/2)=1}}}
This is an equation of an ellipse with vertical major axis.
Ita standard form of equation: {{{(x-h)^2/b^2+(y-k)^2/a^2=1}}}, a>b, (h,k)=coordinates of center.
For given ellipse:
center: (4, -25)
a^2=1475/2
a=√(1475/2)≈27.2 (distance from center to vertices)
b^2=1475/9
b=√(1479/9)≈12.8 (distance from center to co-vertices)
see graph below:
{{{ graph( 300, 300, -80,80, -80,80,(737.5-(737.5*(x-4)^2/163.9))^.5-25,-(737.5-(737.5*(x-4)^2/163.9))^.5-25)}}}